Acute triangulations of polyhedra and R^n

Details Acute triangulations of polyhedra and R^n Acute triangulations of polyhedra and R^n

2009-09-21

arxiv.org/abs/0909.3706v3

Mathematics

Metric Geometry

22 pages, 7 figures

Scientific paper

We study the problem of acute triangulations of convex polyhedra and the space R^n. Here an acute triangulation is a triangulation into simplices whose dihedral angles are acute. We prove that acute triangulations of the n-cube do not exist for n>=4. Further, we prove that acute triangulations of the space R^n do not exist for n>= 5. In the opposite direction, in R^3, we present a construction of an acute triangulation of the cube, the regular octahedron and a non-trivial acute triangulation of the regular tetrahedron. We also prove nonexistence of an acute triangulation of R^4 if all dihedral angles are bounded away from pi/2.

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